| L(s) = 1 | + (−0.392 + 1.35i)2-s + (−0.831 − 0.555i)3-s + (−1.69 − 1.06i)4-s + (−0.00368 − 0.000732i)5-s + (1.08 − 0.911i)6-s + (1.78 − 4.31i)7-s + (2.11 − 1.87i)8-s + (0.382 + 0.923i)9-s + (0.00244 − 0.00471i)10-s + (0.362 + 0.541i)11-s + (0.813 + 1.82i)12-s + (1.81 − 0.361i)13-s + (5.16 + 4.12i)14-s + (0.00265 + 0.00265i)15-s + (1.72 + 3.61i)16-s + (0.841 − 0.841i)17-s + ⋯ |
| L(s) = 1 | + (−0.277 + 0.960i)2-s + (−0.480 − 0.320i)3-s + (−0.845 − 0.533i)4-s + (−0.00164 − 0.000327i)5-s + (0.441 − 0.372i)6-s + (0.676 − 1.63i)7-s + (0.747 − 0.664i)8-s + (0.127 + 0.307i)9-s + (0.000772 − 0.00149i)10-s + (0.109 + 0.163i)11-s + (0.234 + 0.527i)12-s + (0.504 − 0.100i)13-s + (1.38 + 1.10i)14-s + (0.000685 + 0.000685i)15-s + (0.430 + 0.902i)16-s + (0.204 − 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.854335 - 0.0928121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.854335 - 0.0928121i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.392 - 1.35i)T \) |
| 3 | \( 1 + (0.831 + 0.555i)T \) |
| good | 5 | \( 1 + (0.00368 + 0.000732i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.78 + 4.31i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.362 - 0.541i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.81 + 0.361i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-0.841 + 0.841i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.692 + 3.47i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-6.06 + 2.51i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (3.09 - 4.63i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 4.46iT - 31T^{2} \) |
| 37 | \( 1 + (1.11 - 5.62i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-5.96 + 2.46i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.72 + 1.15i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (7.56 - 7.56i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.89 - 4.32i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (14.5 + 2.89i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (-10.7 - 7.17i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-3.18 - 2.12i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-1.66 + 4.01i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.48 - 8.42i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-10.8 - 10.8i)T + 79iT^{2} \) |
| 83 | \( 1 + (-2.26 - 11.4i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-2.32 - 0.961i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 0.263iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84042838433160663967120938879, −11.20819943791180531772911924030, −10.61092537736512967929044893732, −9.441655710151008057897444336691, −8.120856014025286333853046302486, −7.31067553418621401686577719956, −6.51456630611088350123484164256, −5.10608949000334760795878280267, −4.11352991112414613786603255398, −1.01552963625132357154023283267,
1.87945331702783353066272072235, 3.51430526374768008043910565475, 5.00964941082711731248853606811, 5.91478279487771258569060383112, 7.87619203231267301757181548517, 8.858216729642550090131632228056, 9.589256666511640199364072446041, 10.87326154371880784849249956746, 11.54286310770139420885757800007, 12.23689316109478988647150512732
